Stable And Efficient Numerical Schemes For Two-dimensional Maxwell Equations

In this thesis,we are mainly concerned with proposing some efficient and accurate numerical schemes for two-dimensional Maxwell equation.We are also attempted to analyze these schemes theoretically,including stability,energy structure-preserving and convergence.It is suggested that these schemes are unconditionally stable,energy structure-preserving,numerical solution conver-=gence with O(?p+ hq).Some numerical experiments are reported to illustrate the theoretical results.It also verifies the efficiency and accurate.The detailed outline is as follows:In Chapter 1,we mainly introduce the physical background,source and research status of the Maxwell equation at home and abroad as well as some notations and lemmas used in this thesis.Then,we decompose the Maxwell equation into two local-one dimensional problems by splitting methods.In Chapter 2,we first briefly study Wendroff scheme which is often used in numerically solving hyperbolic type equations.For one dimensional convection equation,the Wendroff scheme is unconditionally stable and the order of convergence in space and time are both two order.So,we combine Wendroff scheme to put forward the first order of time Wendroff scheme.To improve the splitting error in the time direction,we use Strang splitting and put forward the two order Wendroff scheme.In order to improve the precision of the spatial direction,the high order compact method is used to discretize the spatial direction.The high order compact scheme of the first order and the second order in time direction and fourth order in space direction is obtained respectively.In Chapter 3,based on the new numerical schemes,we conduct some numerical analysis.Firstly,we analyze the stability by Fourier method.It indicates that they are unconditionally stable.Secondly,by energy method,we know that they preserve the energy structures of the continuous system.They are energy-preserving in lossless media and energy dissipative in lossy media.Finally,it is suggested that the solutions of new schemes are convergent to the exact solution of the corresponding Maxwell equation by energy method and introducing intermediate variables.In Chapter 4,we give the experimental results of the four schemes in lossy media and lossless media,and verify the theoretical analysis.